topological space

topological space (plural topological spaces)

1. (topology, formally) An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets);
   (loosely) the set X.
   • 1964, H. J. Kowalsky, translated by Jay E. Strum, Topological Spaces, Academic Press, page 165:

     In order to obtain "intuitive insight" into special classes of topological spaces we can proceed in several ways, only a few of which will be pursued in this chapter. For instance, we can seek to describe important topological spaces by means of enough of their properties to completely characterize them, up to homeomorphism.

   • 2011, Jonathan A. Barmak, Algebraic Topology of Finite Topological Spaces and Applications, Springer, Lecture Notes in Mathematics 2032, page xi,

     Most of the spaces studied in Algebraic Topology, such as CW-complexes or manifolds, are Hausdorff. In contrast, finite topological spaces are rarely Hausdorff. A topological space with finitely many points, each of which is closed, must be discrete.

   • 2013, D. J. H. Garling, A Course in Mathematical Analysis: Volume 2, Cambridge University Press, page 431:

     If ${\displaystyle (X,\tau )}$ is a topological space, then a cover ${\displaystyle {\mathcal {B}}}$ is open if each ${\displaystyle B\in {\mathcal {B}}}$ is an open set. A topological space ${\displaystyle (X,\tau )}$ is compact if every open cover of ${\displaystyle X}$ has a finite subcover.

• The set ${\displaystyle X}$ may be called the underlying set.
• The collection ${\displaystyle \tau }$ may be called the topology.
• The topological space ${\displaystyle (X,\tau )}$ may be described as the underlying set ${\displaystyle X}$ endowed with the topology ${\displaystyle \tau }$.

• mathematical space, space

• Cauchy space
  • uniform space
    • metric space
• normal space
• pointed space
• regular space
• sober space
  • Hausdorff space
• suspension

• topology
• open set
• underlying set

• Topology on Wikipedia.Wikipedia
• Specialization (pre)order on Wikipedia.Wikipedia
• Spectral space on Wikipedia.Wikipedia
• Topological space on Encyclopedia of Mathematics